Learn how to find the derivative of 1/x^2. The derivative is d/dx[1/x^2].
Below is the graph of f(x) = 1/x^2 and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
The function 1/x^2 requires the use of the Quotient Rule because it is a ratio of two functions. The Power Rule also applies because 1/x^2 can be rewritten as x^(-2), which simplifies the differentiation process.
💡 Why this works: The Quotient Rule helps us differentiate the ratio of two functions, while the Power Rule allows us to handle the negative exponent after re-writing the function.
First, rewrite 1/x^2 as x^(-2). Then, apply the Power Rule to differentiate. The Power Rule states that d/dx[x^n] = n * x^(n-1), so applying it to x^(-2) gives -2 * x^(-3).
💡 Why this works: By using the Power Rule, we find the derivative of x^(-2) is -2 * x^(-3), which is the derivative of 1/x^2.
Simplify the expression -2 * x^(-3) to -2/x^3 to obtain the final derivative. This result shows that the function’s rate of change is negative and decreases as x increases.
💡 Why this works: The simplified form of the derivative confirms the result is correct and provides a clear, interpretable expression for the rate of change of 1/x^2.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
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